Theorem cnvcnv 3486

Description: The double converse of a class strips out all elements that are not ordered pairs.

Assertion

Ref	Expression
cnvcnv	|- `'`'A = (A i^i (V X. V))

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		cnvin 3456	. . 3 |- `'(A i^i (V X. V)) = (`'A i^i `'(V X. V))
2		cnveq 3292	. . 3 |- (`'(A i^i (V X. V)) = (`'A i^i `'(V X. V)) -> `'`'(A i^i (V X. V)) = `'(`'A i^i `'(V X. V)))
3	1, 2	ax-mp 7	. 2 |- `'`'(A i^i (V X. V)) = `'(`'A i^i `'(V X. V))
4		inss2 2231	. . . 4 |- (A i^i (V X. V)) (_ (V X. V)
5		df-rel 3185	. . . 4 |- (Rel (A i^i (V X. V)) <-> (A i^i (V X. V)) (_ (V X. V))
6	4, 5	mpbir 190	. . 3 |- Rel (A i^i (V X. V))
7		dfrel2 3485	. . 3 |- (Rel (A i^i (V X. V)) <-> `'`'(A i^i (V X. V)) = (A i^i (V X. V)))
8	6, 7	mpbi 189	. 2 |- `'`'(A i^i (V X. V)) = (A i^i (V X. V))
9		cnvin 3456	. . 3 |- `'(`'A i^i `'(V X. V)) = (`'`'A i^i `'`'(V X. V))
10		relcnv 3435	. . . . . 6 |- Rel `'`'A
11		df-rel 3185	. . . . . 6 |- (Rel `'`'A <-> `'`'A (_ (V X. V))
12	10, 11	mpbi 189	. . . . 5 |- `'`'A (_ (V X. V)
13		relxp 3255	. . . . . 6 |- Rel (V X. V)
14		dfrel2 3485	. . . . . 6 |- (Rel (V X. V) <-> `'`'(V X. V) = (V X. V))
15	13, 14	mpbi 189	. . . . 5 |- `'`'(V X. V) = (V X. V)
16	12, 15	sseqtr4 2094	. . . 4 |- `'`'A (_ `'`'(V X. V)
17		dfss 2054	. . . 4 |- (`'`'A (_ `'`'(V X. V) <-> `'`'A = (`'`'A i^i `'`'(V X. V)))
18	16, 17	mpbi 189	. . 3 |- `'`'A = (`'`'A i^i `'`'(V X. V))
19	9, 18	eqtr4 1498	. 2 |- `'(`'A i^i `'(V X. V)) = `'`'A
20	3, 8, 19	3eqtr3r 1504	1 |- `'`'A = (A i^i (V X. V))

Syntax hints:   = wceq 956  Vcvv 1811   i^i cin 2046   (_ wss 2047   X. cxp 3168  `'ccnv 3169  Rel wrel 3175

This theorem is referenced by:  cnvcnv2 3487  cnvcnvss 3488  rescnvcnv 3493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186

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